Evaluate the following powers of $i$.
${\left(4i\right)}^{-3}$

• A. $64i^3$
• B. $64i$
• C. $\frac{1}{64}i$
• D. $\frac{1}{64}{i}^2$

Answer: (C)

Verified step by step guidance

1. Rewrite the expression with the negative exponent as a reciprocal: \(\left(4i\right)^{-3} = \frac{1}{\left(4i\right)^3}\).
2. Calculate the cube of the product inside the denominator: \(\left(4i\right)^3 = 4^3 \cdot i^3\).
3. Evaluate the numerical part: \$4^3 = 64$, so the expression becomes \(\frac{1}{64 \cdot i^3}\).
4. Recall the powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = i^2 \cdot i = -1 \cdot i = -i\).
5. Substitute \(i^3\) with \(-i\) in the denominator and simplify the fraction accordingly.